To solve eq:pde, one can parametrize \(u(x)\)  as a neural network, i.e., \(u(x; \theta )\) . This is natural in modern supervised learning, and is called physics-informed neural networks (PINNs) in the numerical PDE community [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. In numerical PDEs, classical methods use finite-difference approximations for all the partial derivatives, whereas the PINN idea directly works with continuous functions that allow partial derivatives to be computed via auto-differentiation. This “mesh-free" nature of PINNs holds the promise for high-precision solutions even for high-dimensional PDEs.
